Solve (600-10x)(5+x)=5500

$$(600-10x)(5+x)=5500$$

Solve for x

$$x=5$$

$$x=50$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 600-10x by 5+x and combine like terms.

$$3000+550x-10x^{2}=5500$$

Subtract 5500 from both sides.

$$3000+550x-10x^{2}-5500=0$$

Subtract 5500 from 3000 to get -2500.

$$-2500+550x-10x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-10x^{2}+550x-2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 550 for b, and -2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-550±\sqrt{550^{2}-4\left(-10\right)\left(-2500\right)}}{2\left(-10\right)}$$

Square 550.

$$x=\frac{-550±\sqrt{302500-4\left(-10\right)\left(-2500\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-550±\sqrt{302500+40\left(-2500\right)}}{2\left(-10\right)}$$

Multiply 40 times -2500.

$$x=\frac{-550±\sqrt{302500-100000}}{2\left(-10\right)}$$

Add 302500 to -100000.

$$x=\frac{-550±\sqrt{202500}}{2\left(-10\right)}$$

Take the square root of 202500.

$$x=\frac{-550±450}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-550±450}{-20}$$

Now solve the equation x=\frac{-550±450}{-20} when ± is plus. Add -550 to 450.

$$x=-\frac{100}{-20}$$

Divide -100 by -20.

$$x=5$$

Now solve the equation x=\frac{-550±450}{-20} when ± is minus. Subtract 450 from -550.

$$x=-\frac{1000}{-20}$$

Divide -1000 by -20.

$$x=50$$

The equation is now solved.

$$x=5 x=50$$

Steps for Completing the Square

Use the distributive property to multiply 600-10x by 5+x and combine like terms.

$$3000+550x-10x^{2}=5500$$

Subtract 3000 from both sides.

$$550x-10x^{2}=5500-3000$$

Subtract 3000 from 5500 to get 2500.

$$550x-10x^{2}=2500$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-10x^{2}+550x=2500$$

Divide both sides by -10.

$$\frac{-10x^{2}+550x}{-10}=\frac{2500}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{550}{-10}x=\frac{2500}{-10}$$

Divide 550 by -10.

$$x^{2}-55x=\frac{2500}{-10}$$

Divide 2500 by -10.

$$x^{2}-55x=-250$$

Divide -55, the coefficient of the x term, by 2 to get -\frac{55}{2}. Then add the square of -\frac{55}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-55x+\left(-\frac{55}{2}\right)^{2}=-250+\left(-\frac{55}{2}\right)^{2}$$

Square -\frac{55}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-55x+\frac{3025}{4}=-250+\frac{3025}{4}$$

Add -250 to \frac{3025}{4}.

$$x^{2}-55x+\frac{3025}{4}=\frac{2025}{4}$$

Factor x^{2}-55x+\frac{3025}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{55}{2}\right)^{2}=\frac{2025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Simplify.

$$x-\frac{55}{2}=\frac{45}{2} x-\frac{55}{2}=-\frac{45}{2}$$

Add \frac{55}{2} to both sides of the equation.

$$x=50 x=5$$